Additive maps preserving operators of inner local spectral radius zero

Let \({\mathcal {B}}(X)\) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space \(X\) . We determine the form of surjective additive maps \(\varphi :{\mathcal {B}}(X)\rightarrow {{\mathcal {B}}(X)}\) which preserve operators of inner local spectral radius zero at points of \(X\) .     

Operator Ideals on Non-commutative Function Spaces

Suppose X and Y are Banach spaces, and \({{\mathcal{I}}}\) , \({{\mathcal{J}}}\) are operator ideals. compact operators). Under what conditions does the inclusion \({\mathcal{I}(X,Y) \subset \mathcal{J}(X,Y)}\) , or the equality \({\mathcal{I}(X,Y)\,=\,\mathcal{J}(X,Y)}\) , hold? We examine this question when \({\mathcal{I}, \mathcal{J}}\) are the ideals of Dunford–Pettis, strictly (co)singular, finitely strictly singular, inessential, or (weakly) compact operators, while X and Y are non-commutative function spaces. Since such spaces are ordered, we also address the same questions for positive parts of such ideals.     

Finite frames,P-frames and basically disconnected frames

Let \({\mathcal{P}}\) be an ideal of closed quotients of a completely regular frame L and \({\mathcal{R}_{\mathcal{P}}(L)}\) the collection of all functions in the ring \({\mathcal{R}(L)}\) whose support belong to \({\mathcal{P}}\) . We show that \({\mathcal{R}(L)}\) is a Noetherian ring if and only if \({\mathcal{R}(L)}\) is an Artinian ring if and only if L is a finite frame. Using this result, we next show that if \({\mathcal{P}}\) is the ideal of all compact closed quotients of L and L is \({\mathcal{P}}\) -continuous, then \({\mathcal{R}_{\mathcal{P}}(L)}\) is a Noetherian ring if and only if L is finite. Moreover, we show that L is a P-frame if and only if each ideal of \({\mathcal{R}(L)}\) is of the form \({\mathcal{R}_{\mathcal{P}}(L)}\) for some choice of \({\mathcal{P}}\) . We furnish equivalent conditions for \({\mathcal{R}_{\mathcal{P}}(L)}\) to be a prime ideal, a free ideal, and an essential ideal of \({\mathcal{R}(L)}\) separately in terms of the cozero elements of L. Finally, we show that L is basically disconnected if and only if \({\mathcal{R}(L)}\) is a coherent ring.     

On the Dirichlet and Neumann Evolution Operators in ℝ d +

We prove some uniform and pointwise gradient estimates for the Dirichlet and the Neumann evolution operators \(G_{\mathcal {D}}(t,s)\) and \(G_{\mathcal {N}}(t,s)\) associated with a class of nonautonomous elliptic operators (t) with unbounded coefficients defined in I× \(\mathbb{R}_{+}\) (where I is a right-halfline or I=?). We also prove the existence and the uniqueness of a tight evolution system of measures \(\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}\) associated with \(G_{\mathcal {N}}(t,s)\) , which turns out to be sub-invariant for \(G_{\mathcal {D}}(t,s)\) , and we study the asymptotic behaviour of the evolution operators \(G_{\mathcal {D}}(t,s)\) and \(G_{\mathcal {N}}(t,s)\) in the L ^p -spaces related to the system \(\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}\) .     

Procrustes Problems and Parseval Quasi-Dual Frames

Parseval frames have particularly useful properties, and in some cases, they can be used to reconstruct signals which were analyzed by a non-Parseval frame. In this paper, we completely describe the degree to which such reconstruction is feasible. Indeed, notice that for fixed frames \({\mathcal{F}}\) and \({\mathcal{X}}\) with synthesis operators F and X, the operator norm of FX ^??I measures the (normalized) worst-case error in the reconstruction of vectors when analyzed with \({\mathcal{X}}\) and synthesized with \({\mathcal{F}}\) . Hence, for any given frame \({\mathcal{F}}\) , we compute explicitly the infimum of the operator norm of FX ^??I, where \({\mathcal{X}}\) is any Parseval frame. The \({\mathcal{X}}\) ’s that minimize this quantity are called Parseval quasi-dual frames of \({\mathcal{F}}\) . Our treatment considers both finite and infinite Parseval quasi-dual frames.     

Categorical group invariants of 3-manifolds

For a given class \({\mathcal{G}}\) of groups, a 3-manifold M is of \({\mathcal{G}}\) -category \({\leq k}\) if it can be covered by k open subsets such that for each path-component W of the subsets the image of its fundamental group \({ \pi_1(W) \rightarrow \pi(M )}\) belongs to \({\mathcal{G}}\) . The smallest number k such that M admits such a covering is the \({\mathcal{G}}\) -category, \({cat_{\mathcal{G}}(M)}\) . If M is closed, it has \({\mathcal{G}}\) -category between 1 and 4. We characterize all closed 3-manifolds of \({\mathcal{G}}\) -category 1, 2, and 3 for various classes \({\mathcal{G}}\) .     

Trace Formulae for Curvature of Jet Bundles over Planar Domains

For a domain \(\varOmega \) in \(\mathbb {C}\) and an operator \(T\) in \({\mathcal {B}}_n(\varOmega )\) , Cowen and Douglas construct a Hermitian holomorphic vector bundle \(E_T\) over \(\varOmega \) corresponding to \(T\) . The Hermitian holomorphic vector bundle \(E_T\) is obtained as a pull-back of the tautological bundle \(S(n,{\mathcal {H}})\) defined over \({\mathcal {G}}r(n,{\mathcal {H}})\) by a nondegenerate holomorphic map \(z\mapsto {\mathrm{ker}}(T-z),\;z\in \varOmega \) . To find the answer to the converse, Cowen and Douglas studied the jet bundle in their foundational paper. The computations in this paper for the curvature of the jet bundle are rather intricate. They have given a set of invariants to determine if two rank \(n\) Hermitian holomorphic vector bundle are equivalent. These invariants are complicated and not easy to compute. It is natural to expect that the equivalence of Hermitian holomorphic jet bundles should be easier to characterize. In fact, in the case of the Hermitian holomorphic jet bundle \({\mathcal {J}}_k({\mathcal {L}}_f)\) , we have shown that the curvature of the line bundle \({\mathcal {L}}_f\) completely determines the class of \({\mathcal {J}}_k({\mathcal {L}}_f)\) . In case of rank \(n\) Hermitian holomorphic vector bundle \(E_f\) , We have calculated the curvature of jet bundle \({\mathcal {J}}_k(E_f)\) and also obtained a trace formula for jet bundle \({\mathcal {J}}_k(E_f)\) .     

Transformations of polar Grassmannians preserving certain intersecting relations

Let Π be a polar space of rank n≥3. Denote by \({\mathcal{G}}_{k}(\varPi)\) the polar Grassmannian formed by singular subspaces of Π whose projective dimension is equal to k. Suppose that k is an integer not greater than n?2 and consider the relation \({\mathfrak{R}}_{i,j}\) , 0≤i≤j≤k+1, formed by all pairs \((X,Y)\in{\mathcal{G}}_{k}(\varPi)\times{\mathcal{G}}_{k}(\varPi)\) such that dim _p (X ^⊥∩Y)=k?i and dim _p (X∩Y)=k?j (X ^⊥ consists of all points of Π collinear to every point of X). We show that every bijective transformation of \({\mathcal{G}}_{k}(\varPi)\) preserving \({\mathfrak{R}}_{1,1}\) is induced by an automorphism of Π, except the case where Π is a polar space of type D _n with lines containing precisely three points. If k=n?t?1, where t is an integer satisfying n≥2t≥4, we show that every bijective transformation of \({\mathcal{G}}_{k}(\varPi)\) preserving \({\mathfrak{R}}_{0,t}\) is induced by an automorphism of Π.     

Approximating minimum-cost edge-covers of crossing biset-families

Part of this paper appeared in the preliminary version [16]. An ordered pair ? = (S, S ₊) of subsets of a groundset V is called a biset if S ? S+; (V S ⁺;V S) is the co-biset of ?. Two bisets \(\hat X,\hat Y\) intersect if ^X X ∩ Y ≠ \(\not 0\) and cross if both X ∩ Y \(\not 0\) and X ⁺ ∪ Y ⁺ ≠= V. The intersection and the union of two bisets \(\hat X,\hat Y\) are defined by \(\hat X \cap \hat Y = (X \cap Y,X^ + \cap Y^ + )\) and \(\hat X \cup \hat Y = (X \cup Y,X^ + \cup Y^ + )\) . A biset-family \(\mathcal{F}\) is crossing (intersecting) if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) that cross (intersect). A directed edge covers a biset ? if it goes from S to V S ⁺. We consider the problem of covering a crossing biset-family \(\mathcal{F}\) by a minimum-cost set of directed edges. While for intersecting \(\mathcal{F}\) , a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing \(\mathcal{F}\) is not yet understood, as it includes several NP-hard problems, for which a poly-logarithmic approximation was discovered only recently or is not known. Let us say that a biset-family \(\mathcal{F}\) is k-regular if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) with |V (X∪Y)≥k+1 that intersect. In this paper we obtain an O(log |V|)-approximation algorithm for arbitrary crossing \(\mathcal{F}\) if in addition both \(\mathcal{F}\) and the family of co-bisets of \(\mathcal{F}\) are k-regular, our ratios are: \(O\left( {\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) , and \(O\left( {\frac{{|V|}} {{|V| - k}}\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) . Using these generic algorithms, we derive for some network design problems the following approximation ratios: \(O\left( {\log k \cdot \log \tfrac{n} {{n - k}}} \right) \) for k-Connected Subgraph, and O(logk) \(\min \{ \tfrac{n} {{n - k}}\log \tfrac{n} {{n - k}},\log k\} \) for Subset k-Connected Subgraph when all edges with positive cost have their endnodes in the subset.     

Badly approximable points on planar curves and a problem of Davenport

Let \({\mathcal {C}}\) be two times continuously differentiable curve in \({\mathbb {R}}^2\) with at least one point at which the curvature is non-zero. For any \(i,j \geqslant 0\) with \(i+j =1\) , let \({\mathbf {Bad}}(i,j)\) denote the set of points \((x,y) \in {\mathbb {R}}^2\) for which \( \max \{ \Vert qx\Vert ^{1/i}, \, \Vert qy\Vert ^{1/j} \} > c/q \) for all \( q \in {\mathbb {N}}\) . Here \(c = c(x,y)\) is a positive constant. Our main result implies that any finite intersection of such sets with \({\mathcal {C}}\) has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.     

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

Forbidden Triples Containing a Complete Graph and a Complete Bipartite Graph of Small Order

For a graph G and a set \({\mathcal{F}}\) of connected graphs, G is said be \({\mathcal{F}}\) -free if G does not contain any member of \({\mathcal{F}}\) as an induced subgraph. We let \({\mathcal{G} _{3}(\mathcal{F})}\) denote the set of all 3-connected \({\mathcal{F}}\) -free graphs. This paper is concerned with sets \({\mathcal{F}}\) of connected graphs such that \({\mathcal{F}}\) contains no star, \({|\mathcal{F}|=3}\) and \({\mathcal{G}_{3}(\mathcal{F})}\) is finite. Among other results, we show that for a connected graph T( ≠ K ₁) which is not a star, \({\mathcal{G}_{3}(\{K_{4},K_{2,2},T\})}\) is finite if and only if T is a path of order at most 6.     

Spectra of tensor triangulated categories over category algebras

Let \({\mathcal{C}}\) be a finite EI category and k be a field. We consider the category algebra \({k\mathcal{C}}\) . Suppose \({\sf{K}(\mathcal{C})=\sf{D}^b(k \mathcal{C}-\sf{mod})}\) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category, and we compute its spectrum in the sense of Balmer. When \({\mathcal{C}=G \propto \mathcal{P}}\) is a finite transporter category, the category algebra becomes Gorenstein, so we can define the stable module category \({\underline{\sf{CM}} k(G \propto \mathcal{P})}\) , of maximal Cohen–Macaulay modules, as a quotient category of \({{\sf{K}}(G \propto \mathcal{P})}\) . Since \({\underline{\sf{CM}} k(G\propto\mathcal{P})}\) is also tensor triangulated, we compute its spectrum as well. These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories.     

Complemented modular lattices with involution and orthogonal geometry

With each orthogeometry (P, ⊥) we associate ${{\mathbb {L}}(P, \bot)}$ , a complemented modular lattice with involution (CMIL), consisting of all subspaces X and X ^⊥ such that dim X < ?₀, and we study its rôle in decompositions of (P, ⊥) as directed (resp., disjoint) union. We also establish a 1–1 correspondence between ?-varieties ${\mathcal {V}}$ of CMILs with ${\mathcal {V}}$ generated by its finite dimensional members and ‘quasivarieties’ ${\mathcal {G}}$ of orthogeometries: ${\mathcal {V}}$ consists of the CMILs representable within some geometry from ${\mathcal {G}}$ and ${\mathcal {G}}$ of the (P, ⊥) with ${{\mathbb {L}}(P, \bot) \in {\mathcal {V}}}$ . Here, ${\mathcal {V}}$ is recursively axiomatizable if and only if so is ${\mathcal {G}}$ . It follows that the equational theory of ${\mathcal {V}}$ is decidable provided that the equational theories of the ${\{{\mathbb {L}}(P, \bot)\, |\, (P, \bot) \in \mathcal {G}, {\rm{dim}} P = n\}}$ are uniformly decidable.     

Valuation Extensions of Algebras Defined by Monic Gr?bner Bases

Let K be a field, $\mathcal {O}_v$ a valuation ring of K associated to a valuation v: K → Γ?∪?{?∞?}, and m _v the unique maximal ideal of $\mathcal {O}_v$ . Consider an ideal $\mathcal {I}$ of the free K-algebra $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ on X ₁,...,X _n . If ${\cal I}$ is generated by a subset $\mathcal {G}\subset{\cal O}_v\langle X\rangle$ which is a monic Gr?bner basis of ${\cal I}$ in $K\langle X\rangle$ , where $\mathcal {O}_v\langle X\rangle =\mathcal{O}_v\langle X_1,...,X_n\rangle$ is the free $\mathcal{O}_v$ -algebra on X ₁,...,X _n , then the valuation v induces naturally an exhaustive and separated Γ-filtration F ^v A for the K-algebra $A=K\langle X\rangle /\mathcal {I}$ , and moreover $\mathcal{I}\cap\mathcal{O}_v\langle X\rangle =\langle\mathcal{G}\rangle$ holds in $\mathcal{O}_v\langle X\rangle$ ; it follows that, if furthermore $\mathcal{G}\not\subset {\bf m}_v{O}_v\langle X\rangle$ and $k\langle X\rangle /\langle\overline{\mathcal G}\rangle$ is a domain, where $k=\mathcal{O}_v/{\bf m}_v$ is the residue field of $\mathcal{O}_v$ , $k\langle X\rangle =k\langle X_1,...,X_n\rangle$ is the free k-algebra on X ₁,...,X _n , and $\overline{\mathcal G}$ is the image of $\mathcal{G}$ under the canonical epimorphism $\mathcal{O}_v\langle X\rangle\rightarrow k\langle X\rangle$ , then F ^v A determines a valuation function A → Γ?∪?{?∞?}, and thereby v extends naturally to a valuation function on the (skew-)field Δ of fractions of A provided Δ exists.     

Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, \mathcal{A})}$ of cardinality ${|A| \geq 2}$ and infinite set X of cardinality ${|X| \geq | \mathcal{A}|}$ , the X-th power ${(A^{X}, \mathcal{A}^{X})}$ of the algebra ${(A, \mathcal{A})}$ contains a free subset ${\mathcal{F} \subset A^{X}}$ of cardinality ${|\mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${\mathcal{I} \subset \mathcal{P}(X)}$ of cardinality ${|\mathcal{I}| = |\mathcal{P}(X)|}$ in the Boolean algebra ${\mathcal{P}(X)}$ of subsets of an infinite set X.     

Closed Range Property for Holomorphic Semi-Fredholm Functions

Given Banach spaces X and Y, we show that, for each operator-valued analytic map ${\alpha \in \mathcal O (D,\mathcal L(Y,X))}$ satisfying the finiteness condition ${\dim (X/\alpha (z)Y) < \infty}$ pointwise on an open set D in ${\mathbb {C}^n}$ , the induced multiplication operator ${\mathcal O(U,Y) \stackrel{\alpha}{\longrightarrow} \mathcal O (U,X)}$ has closed range on each Stein open set ${U \subset D}$ . As an application we deduce that the generalized range ${{\rm R}^{\infty}(T) = \bigcap_{k \geq 1}\sum_{| \alpha | = k} T^{\alpha}X}$ of a commuting multioperator ${T \in \mathcal L(X)^n}$ with ${\dim(X/\sum_{i=1}^n T_iX) < \infty}$ can be represented as a suitable spectral subspace.     

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$ , where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$ -rational points on ${\mathcal X}$ and E is a divisor on ${\mathcal X}$ . If ${\deg(E)\geq 2g+1}$ where g is the genus of ${\mathcal X}$ , then there is an embedding of ${\mathcal X}$ onto ${\mathcal Y}$ in the projective space of the linear series of the divisor E. Moreover, if ${\deg(E)\geq 2g+2}$ , then ${I(\mathcal Y)}$ , the vanishing ideal of ${\mathcal Y}$ , is generated by ${I_2(\mathcal Y)}$ , the homogeneous elements of degree two in ${I(\mathcal Y)}$ . If ${n >2 \deg(E)}$ , then ${I_2(\mathcal Y)=I_2(\mathcal Q)}$ , where ${\mathcal Q}$ is the image of ${\mathcal P}$ under the map from ${\mathcal X}$ to ${\mathcal Y}$ . These three results imply that, if ${2g+2\leq m < \frac{1}{2}n}$ , an AG representation ${(\mathcal Y, \mathcal Q, F)}$ of the code ${\mathcal C}$ can be obtained just using a generator matrix of ${\mathcal C}$ where ${\mathcal Y}$ is a normal curve in ${\mathbb{P}^{m-g}}$ which is the intersection of quadrics. This fact gives us some clues for breaking McEliece cryptosystem based on AG codes provided that we have an efficient procedure for computing and decoding the representation obtained.     

Special values of anticyclotomic L-functions modulo λ

The purpose of this article is to generalize some results of Vatsal on the special values of Rankin–Selberg L-functions in an anticyclotomic \({\mathbb{Z}_{p}}\) -extension. Let g be a cuspidal Hilbert modular newform of parallel weight \({(2,\ldots,2)}\) and level \({\mathcal{N}}\) over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant \({\mathcal{D}}\) . We study the l-adic valuation of the special values \({L(g,\chi,\frac{1}{2})}\) as \({\chi}\) varies over the ring class characters of K of \({\mathcal{P}}\) -power conductor, for some fixed prime ideal \({\mathcal{P}}\) . We prove our results under the only assumption that the prime to \({\mathcal{P}}\) part of \({\mathcal{N}}\) is relatively prime to \({\mathcal{D}}\) .